This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
When I started writing about my favorite topological spaces, I knew I’d have to confront open sets someday. In topology, it’s not enough to define a space by saying what points are in it. Every topological space comes with baggage: open sets. There are only a few rules: the total space and the empty set both have to be open sets, the intersection of any finite number of open sets has to be open, and the union, or combination, of any collection of open sets has to be an open set as well.
In some sense, all open sets are generalizations of an open interval like (0,1) on the number line. (If math notation is a distant memory, the parentheses here mean we include all the numbers greater than 0 and less than 1 but not 0 and 1 themselves. If we wanted to include the endpoints, we would write [0,1] and call it a closed interval.) In order for us to satisfy all the requirements above, we also declare that the empty set is an open set, as are unbounded intervals like the set of all numbers greater than 3, which we would write (3,∞), and that collections of open intervals are open.
But we could make different decisions about what sets are open and end up with a different topological space. One alternative to the standard topology is called the lower limit topology. In this topology, open sets are half-open intervals: [0,1), for example. The only difference is that we now include the left endpoint. When we analyze the rule that unions of open sets must be open, we discover that when we define sets like [0,1) to be open, we get sets like (0,1) as well. Thus the lower limit topology is definitely different from the standard topology: everything that is open in the standard topology is open in the lower limit topology, but the lower limit topology also has other open sets the standard topology doesn't.
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Why should we care about open sets? Far be it from me to tell you how to feel, but mathematicians care about open sets because they allow us to determine the properties of functions on very abstract spaces.
Discontinuity. Image: Alan Joyce, via Flickr.
The most important property of functions between spaces is continuity. Continuity is intuitively obvious for spaces like the number line (with the usual topology): a function is continuous if you can draw its graph without picking up your pencil. There are no big jumps anywhere.
The intuitive definition of continuity requires us to be able to measure the distance between points, but sometimes mathematicians want to be able to define continuity for functions between spaces that don’t necessarily have a built-in distance function. The topological definition of continuity only requires open sets. A function is continuous if the preimage of every open set is open. The preimage of a set is just the collection of points that are mapped to that set under the function. For example, if the function is f(x)=x2, the preimage of 1 is the union of the points 1 and -1.
To see the importance of open sets to continuity, let’s go back to the number line. The standard topology with open intervals as the open sets is just the beginning. With that topology, continuous functions are exactly what we expect: no big jumps. The easiest alternative topology is the indiscrete topology. In that one, we decide that the only open sets are the entire line and the empty set. After all, we're busy people, and we've got other important things to do. Now let’s think about a function from the number line to itself with the indiscrete topology. No matter what the function is, we only care about the preimage of the entire line because that's the only nontrivial open set. If the function is defined on the entire line, then the preimage of the line is the entire line, an open set. Therefore any function is continuous if the target space has the indiscrete topology.
The discrete topology, on the other hand, goes small. In this one, every individual point is an open set. The rules of open sets then declare that every combination of points is an open set. Although this is in some way the opposite of the indiscrete topology, it also has the effect of drastically simplifying the process of figuring out whether a function is continuous. If you have a function from the number line with the discrete topology, it doesn’t matter what the function or even target space is. The function will be continuous because there are no non-open sets in the discrete topology.
When students first encounter beginning topology, they are sometimes disoriented by the power they wield. Mathematical spaces don't always come equipped with their open sets, so we get to define them by fiat. Of course, some open sets are more useful than others, but in the end, we are masters of our topological destinies.
